Tournament quasirandomness from local counting

TL;DR

This paper explores how local and global counts of specific tournaments relate to quasirandomness, revealing that only transitive tournaments are globally forcing and identifying non-transitive locally forcing tournaments.

Contribution

It demonstrates that in tournaments, only transitive structures are globally forcing, and constructs infinite families of non-transitive locally forcing tournaments, contrasting with graph hypergraph analogues.

Findings

Global counts of transitive tournaments imply quasirandomness for large h.

Not all tournaments are locally forcing, with many non-transitive examples.

Locally forcing tournaments must be strongly quasirandom.

Abstract

A well-known theorem of Chung and Graham states that if $h\geq 4$ then a tournament $T$ is quasirandom if and only if $T$ contains each $h$-vertex tournament the "correct number" of times as a subtournament. In this paper we investigate the relationship between quasirandomness of $T$ and the count of a single $h$-vertex tournament $H$ in $T$. We consider two types of counts, the global one and the local one. We first observe that if $T$ has the correct global count of $H$ and $h \geq 7$ then quasirandomness of $T$ is only forced if $H$ is transitive. The next natural question when studying quasirandom objects asks whether possessing the correct local counts of $H$ is enough to force quasirandomness of $T$. A tournament $H$ is said to be locally forcing if it has this property. Variants of the local forcing problem have been studied before in both the graph and hypergraph settings. Perhaps the closest analogue of our problem was considered by Simonovits and S\'os who looked at whether having "correct counts" of a fixed graph $H$ as an induced subgraph of $G$ implies $G$ must be quasirandom, in an appropriate sense. They proved that this is indeed the case when $H$ is regular and conjectured that it holds for all $H$ (except the path on 3 vertices). Contrary to the Simonovits-S\'os conjecture, in the tournament setting we prove that a constant proportion of all tournaments are not locally forcing. In fact, any locally forcing tournament must itself be strongly quasirandom. On the other hand, unlike the global forcing case, we construct infinite families of non-transitive locally forcing tournaments.